The Weak Vop\v{e}nka Principle for definable classes of structures

Abstract

We give a level-by-level analysis of the Weak Vop\v{e}nka Principle for definable classes of relational structures (WVP), in accordance with the complexity of their definition, and we determine the large-cardinal strength of each level. Thus, in particular we show that WVP for $\Sigma_2$-definable classes is equivalent to the existence of a strong cardinal. The main theorem shows, more generally, that WVP for $\Sigma_n$-definable classes is equivalent to the existence of a $\Sigma_n$-strong cardinal. Hence, WVP is equivalent to the existence of a $\Sigma_n$-strong cardinal, all $n <\omega$.